If I understand Hawking's last paper, "A smooth exit from eternal inflation?" (2018-04-27), correctly it implies that the Big Bang generated more universes than our own.
This means that to locate a point in all of the universes we would need a 5-D Coordinate system $\left(x,y,z,t,u\right)$.
Als...
Cantor's reason for devising set theory, i.e., for his "painstaking and hardly rewarding business of investigating point sets", was its application to reality.
"I refer you to what I have found in Math. Annalen Vol. XX pp. 118-121, that in the space filled with body matter (since I assume the...
Let $A = \{(x,y) \in R^{2} : x^{n} + y^{n}=1 \}$. I need to prove that A is bounded implies $n$ is even and vice-versa. The topology is not mentioned in the question, so we take it to be the usual topology on $R^2$. Can someone drop me a hint please.
I have no idea how a systematic proof can be done, but my guess is that when n is odd, you have a sum of potentially positive and negative numbers, and these allow the (x,y) to go unbounded as as long their magnitudes matches in some way, they cancel into 1
This has nothing to do with differentiation per se. It is a simple consequence of the following basic fact about convergence in ${\mathbb R}^n$:
$${\bf x}\to{\bf 0}\qquad\Longleftrightarrow\qquad x_i\to0\quad (1\leq i\leq n)\ .$$
idk, the whole Minkowksi stuff we did in ANT felt a little out of place. You develop this whole theory which isn't related to anything else in the lecture to prove two theorems: Minkowski bound and Dirichlet unit theorem. Granted, these are important theorems, but you could basically forget everything about Minkowski theory/geometry of numbers after that
Or maybe I'll find another interesting topic, I'll think about it next week, I'm busy preparing a presentation on the cardinalities of ultraproducts for this Thursday atm
https://math.stackexchange.com/questions/1550962/polar-decomposition-of-bounded-normal-operator-on-hilbert-space in the accepted answer, how do we know $\mathcal{R}(|T|)^{\perp} =\mathcal{N}(|T|)$?
@Mathei the usual stuff I think, number fields, discriminant, trace and integral (or integer?) basis, dedekind rings and the factorization of ideals, the ideal class group, the Dirichlet unit's theorem and the Minkowski bound
We followed those notes from chapter $6$ till the end, they're in Italian but you can probably make sense of the index without issues
@AlessandroCodenotti it seems that there is no section on cyclotomic fields. That's a pretty big topic, but there's also some rather elementary stuff you can do
Ah, btw @Mathei I tagged you in a ring theory question a week or so ago but you probably didn't see it, do you want to hear it? It's mostly a sanity check
some stuff for cyclotomic fields turns out to be nice and simple: the ring of integers and the splitting of rational primes have simple descriptions (and you don't need more thatn that + a bit of Galois theory for the quadratic reciprocity proof). Other stuff is a bit deeper. The whole field of Iwasawa theory basically started as a way to understand class numbers of cyclotomic fields
The arguments, as far as I understand is, that the order relations of the generators define the Coxeter graph. The Coxeter graph relates to the Coxeter matrix and up to isomorphism there is a one-to-one correspondence between a Coxeter matrix and a Coxeter system.
So the map $\varphi :R/(I\cap J)\to R/I\times R/J$ defined by $\varphi(x+I\cap J)=(x+I,x+J)$ is an isomorphism iff $I$ and $J$ are coprime (one direction is CRT, the other is not hard to prove)
So I wanted to know if $R/(I\cap J)$ and $R/I\times R/J$ can still be isomorphic when $I$ and $J$ are not coprime, just through a different map and I think I have an example
$R=\prod\limits_{i=1}^\infty\Bbb Z$, $I$ is generated by $(1,0,\cdots)$, $J$ is generated by $(0,1,\cdots)$, then $R/(I\cap J)$, $R/I$, $R/J$ and $R\times R$ should all be isomorphic to $R$, does this look right?
hmm, it should be in most books on algebraic number theory. There's a section in chapter 1 of Neukirch, but he uses conductors for the proof that $\mathcal O_{\Bbb Q(\zeta_n)}=\Bbb Z[\zeta_n]$, not sure if you did that
I don't know what those are, but actually now that I think of it there was a series of exercises about cylotomic extensions in a pset which gets very close to proving that $\mathcal O_{\Bbb Q(\zeta_n)}=\Bbb Z[\zeta_n]$. I should check the details
it's also in Pierre Samuel's "algebraic theory of numbers". That's a really small book and the first half is just algebraic background you already know, so you could read up everything else you might need there quickly
@AlessandroCodenotti it's impossible for Noetherian rings from my gut feeling and I have some ideas what might work to show that, but I'm a bit too busy right now to work it out
@LeakyNun , i am learning complex function's differentiablilty, but I want to confirm this: complex function differentiable iff Cauchy Riemann equations hold.
As I said, CRE are not so useful to find the derivative. You use it to prove the derivative exists, and then you use other theorems to actually find the derivative
Or to prove that the function isn't in fact differentiable, since it's an iff
1)x is a random variable defined by the number of the girls who stand in the last 5 places. what is p(x=i) for i=0,1,...,5 ? my solution: $\frac{\binom{7}{i}\binom{13}{5-i}}{\binom{20}{5}}\:5!$
@Tobias ok, then you should know that sometimes not everything is optimal, i.e. people don't get the needed support from their advisor to be productive
@BlueLemon Ohh, my question was not meant to be a criticism. It was meant to get an understanding of how Shaun's day looks, and what might be causing the burnout.
1 hour spent on research, 45 minutes making coffee, 3 hours on MSE, an hour grading, another hour teaching or in office hours, another 30 minutes making coffee, 30 minutes commuting (each way!), another 45 minutes making coffee
@MatheinBoulomenos I have been trying to work out an example. And I have been continuously doing things wrong, even though I have done these sorts of calculations tons of times before a few years back.
Now I have finally gotten my head around all the details, and I realized that I needed a bigger example or it would just be trivial in the setting I was looking at
as a first or second year PhD student, that would be a difficult but not undoable course load
(four graduate level courses in a quarter or semester is a lot, but can be done if you never sleep; the seminars are a timesink, but shouldn't really require that much extra work)
Cantor's reason for devising set theory, i.e., for his "painstaking and hardly rewarding business of investigating point sets", was its application to reality.
"I refer you to what I have found in Math. Annalen Vol. XX pp. 118-121, that in the space filled with body matter (since I assume the...
This has nothing to do with differentiation per se. It is a simple consequence of the following basic fact about convergence in ${\mathbb R}^n$:
$${\bf x}\to{\bf 0}\qquad\Longleftrightarrow\qquad x_i\to0\quad (1\leq i\leq n)\ .$$
